Special Session 110: Evolution Equations with Applications to Control, Mathematical Modeling and Mechanics

A Class of Nonlinear Evolution Equations on Banach Spaces Driven by Finitely Additive Measures and its Optimal Control

Nasir U. Ahmed School of Information Technology and Engineering (SITE), University of Ottawa Canada Co-Author(s):

Abstract: In this paper we consider the following class of nonlinear evolution equations on a Banach space $X$ driven by finitely additive measures as controls $$dx(t) = Ax(t)dt + F (t, x(t))dt + \int_U G(t, x(t), \xi) \mu (dt \times d\xi), $$ where $t \in I = [0, T ]$, $x(0) = x_0$. Here $A$ is the infinitesimal generator of a $C_0$-semigroup on $X$, $F$ is a nonlinear map locally Lipschitz on $X$ having linear growth and $\mu \in {\cal M}_{bfa}(I \times U)$ (the Banach space of bounded finitely additive measures on $I \times U$) is the control with $U$ being a Polish space. This model covers a large class of nonlinear systems driven by measures. In particular,it generalizes the class of classical impulsive systems found in the literature. We prove existence of optimal controls (Measures) and develop Necessary Conditions of Optimality. Also, we present an algorithm and prove a convergence theorem whereby one can construct the optimal control. Further, we present two examples from finite dimensional systems driven by discrete measures (1) Ecological system (2) Geosynchronous Satellite.

Migrations in epidemiological context-a multiscale point of view

Jacek Banasiak Lodz University of Technology/University of Pretoria Poland Co-Author(s):

Abstract: Migrations have been a feature of human life since the dawn of humankind. Usually, they are due to external factors such as environmental changes, natural disasters, economic challenges, or wars. They are instrumental in spreading conditions such as epidemic status, prevalent in particular subpopulations, to the global level. Thus, migration models should be coupled with other processes affecting them. In this talk, we focus on metapopulation models coupling infections with migrations and consider cases when migrations occur either at a much lower or much faster rate than the other processes, leading to, respectively, regularly or singularly perturbed systems of differential equations. Using recent results on uniform-in-time asymptotics, we show how to reduce such models to simpler ones without losing salient features of the original dynamics.

Network models for infection dynamics

Adam Bloch Institute of Mathematics, Lodz University of Technology Poland Co-Author(s):    Jacek Banasiak, Katarzyna Szyma{\`n}ska-D{\k e}bowska

Abstract: In the talk we present general framework for dynamical systems on networks represented by metric graphs and discuss first steps into application of these models to modelling of infection spread, also in the context of armed conflicts. The research is partially supported by the polish National Center of Science grant 2023/05/Y/ST6/00263 Modeling and Forecasting of Infection Spread in War and Post War Settings Using Epidemiological, Behavioral and Genomic Surveillance Data.

A priori estimates for anisotropic systems

Maria-Magdalena Boureanu University of Craiova, Romania Romania Co-Author(s):    Alejandro Velez-Santiago

Abstract: We study anisotropic systems with variable exponents involving Leray-Lions type operators not only in the interior of the domain, but also on the boundary. To justify the study of such general systems, we provide multiple examples of operators satisfying our hypotheses. Our goal is to deduce the existence, the uniqueness, and the global regularity of the weak solution. All these results were recently obtained in a joint work with Alejandro V\`elez\,-\,Santiago.

Set-valued Young integrals and their properties

Mariusz Michta University of Zielona Gora Poland Co-Author(s):    Jerzy Motyl

Abstract: In the talk we present new types of set-valued integrals which establish a generalization of a single-valued Young integral. In a single-valued case the Young integral has been used in a wide range of applications, in particular, one can consider stochastic integration and stochastic equations concerning non-semimartingale integrators such as the Mandelbrot fractional Brownian motion which has Holder continuous sample paths. Thus it seems reasonable to investigate Young-type integrals for multivalued functions and their applications to differential inclusions driven by such new types of integrals. In the presentation, we shall establish properties of different types of set-valued Young integrals for classes of set-valued functions being Holder continuous or with bounded p-variations. These properties are crucial in the studies of different types of Young differential inclusions. $$ $$ References $$ $$ 1. M. Michta, J. Motyl, Selection properties and set-valued Young integrals of set-valued functions, Results Math. 75, 164 (2020). $$ $$ 2. M. Michta, J. Motyl, Set-valued functions of bounded generalized variation and set-valued Young integrals, J. Theor. Probab. 35 (2022), 528-549. $$ $$ 3. M. Michta, J. Motyl, Solution sets for Young differential inclusions, Qual. Theory Dyn. Syst. 22, 132 (2023). $$ $$ 4. M. Michta, J. Motyl, Properties of set-valued Young integrals and Young differential inclusions generated by sets of Holder functions, Nonlinear Differ. Equ. Appl. 31, article no 70 (2024).

Constrained quasi-variational-hemivariational inequalities with applictions

Stanislaw Migorski Jagiellonian University in Krakow, Faculty of Mathematics and Computer Science, Chair of Optimization and Control Poland Co-Author(s):

Abstract: In the talk a class of elliptic quasi-variational-hemivariational inequalities with constraints is examined. The existence of solutions and compactness of the solution set is proved. The upper semicontinuity property of the solution set with respect to a parameter appearing in the data is also established. Then, the results are applied to the stationary incompressible Navier-Stokes equation with mixed multivalued boundary conditions which model a generalized Newtonian fluid of Bingham-type. The corresponding boundary value problem involves a nonmonotone version of the slip condition of frictional type described by the Clarke subgradient law with a locally Lipschitz potential and an implicit obstacle constraint set. Finally, within the framework of optimal control, a double minimization problem for the fluid model is studied and the existence of its solution is established.

Solution sets of Young differential inclusions

Jerzy Motyl University of Zielona Gora Poland Co-Author(s):    Mariusz Michta

Abstract: In the talk we investigate the problem of the existence and properties of solution sets of two types of differential inclusions with Aumann and Young set-valued integrals, presented in their integral form $$x(t)-x(s)\in \int_{s}^{t}F(\tau ,x(\tau ))d\tau +\int_{s}^{t}cl_{\mathcal C}({\mathcal H}\circ x)(\tau)dg(\tau ), \;\;x( 0) =x_0$$ and $$x(t)-x(s)\in \int_{s}^{t}F(\tau ,x(\tau ))d\tau+ \int_{s}^{t}G(\tau ,x(\tau ))dg(\tau ) ,\;\; x(0)=x_0.$$ The existence of solutions, boundedness, compactness of the set of solutions and continuous dependence type results are considered. The obtained results are applied to some optimality problems driven by Young differential inclusions. The properties of optimal solutions and their reachable sets are discussed. These inclusions contain as a particular case set-valued stochastic inclusions with respect to fractional Brownian motion (fBm), and therefore, their properties are crucial for investigation the properties of solutions of (fBm) stochastic differential inclusions. The controlled cancer tumor growth model under the influence of drugs will be presented also. $$ $$ References $$ $$ [1] M. Michta, J. Motyl, Selection Properties and Set-Valued Young Integrals of Set-Valued Functions, Results Math. {\bf 75}, 164 (2020). $$ $$ [2] M. Michta, J. Motyl, Set-Valued Functions of Bounded Generalized Variation and Set-Valued Young Integrals, J. Theor. Probab. {\bf 35}, 528--549 (2022). $$ $$ [3] M. Michta, J. Motyl, Solution Sets for Young Differential Inclusions, Qual. Theory Dyn. Syst. {\bf 22}, 132 (2023). $$ $$ [4] M. Michta, J. Motyl, Properties of set-valued Young integrals and Young differential inclusions generated by sets of H\older functions, Nonlinear Differ. Equ. Appl., Vol. 31, art. no. 70, 1--33, published online June 4, 2024 https://doi.org/10.1007/s00030-024-00963-2

Finite-time synchronization for fuzzy shunting inhibitory cellular neural networks

Zhangir Nuriyev Nazarbayev University Kazakhstan Co-Author(s):    Alfarabi Issakhanov, Jurgen Kurths, Ardak Kashkynbayev

Abstract: Finite-time synchronization (FTS) is a critical problem in the study of neural networks. The primary objective of this study was to construct feedback controllers for various models based on fuzzy shunting inhibitory cellular neural networks (FSICNNs) and find sufficient conditions for the solutions of those systems to reach synchronization in finite time. In particular, by imposing global assumptions of Lipschitz continuous and bounded activation functions, we prove the existence of FTS for three FSICNN models. In general, we consecutively explore models of regular delayed FSICNNs and then consider them in the presence of either inertial or diffusion terms. Using criteria derived by means of the maximum-value approach in its different forms, we give an upper bound of the time up to which synchronization is guaranteed to occur in all three FSICNN models. These results are supported by 2D and 3D computer simulations and two respective numerical examples for $2\times 2$ and $2\times 3$ cases, which show the behaviour of the solutions and errors under different initial conditions of FSICNNs in the presence and absence of designed controllers.

Time-dependent variational-hemivariational inequalities with applications

Anna Ochal Jagiellonian University Poland Co-Author(s):    Anna Kulig (Jagiellonian University in Krakow, Poland)

Abstract: In the talk a class of time-dependent variational-hemivariational inequalities is studied. The existence and uniqueness of a solution are proved. Then, the convergence result for a penalized form of the problem is provided. Finally, the results are applied to a quasistatic frictional contact problem for locking materials.The contact is modeled with friction and a nonsmooth multivalued interface law. The law involves unilateral constraints and subdifferential conditions.

Geometric Approach to Stability of Sets in Differential Inclusions with Maximally Monotone Operators

Hassan Saoud Gulf University for Science and Technology Kuwait Co-Author(s):    Minh Dao, Michel Thera

Abstract: Stability analysis of differential inclusions governed by maximally monotone operators presents significant challenges, especially when focusing on sets rather than isolated equilibria. Traditional methods often require explicit solutions or specific assumptions that may not be feasible. This talk addresses these issues by exploring pointwise asymptotic stability (PAS) and semistability. The approach involves splitting the operator into a convex upper-semicontinuous (CUSCO) mapping and a normal cone, simplifying the problem and allowing for a thorough examination of stability. By using nonsmooth Lyapunov pairs and proximal analysis, this method avoids many traditional assumptions, making the results more widely applicable. This framework extends stability analysis to a broader range of dynamic systems, even when explicit solutions are not available.

Modeling the spread of infection during war

Katarzyna Szymanska-Debowska Institute of Mathematics Lodz University of Technology Poland Co-Author(s):

Abstract: We consider a deterministic model to study the impact of war on the dynamics of viral disease transmission. The starting point is the classical susceptible-infectious-recovered model, which can be enriched with additional classes, for example: contaminated environment, hospitalized persons, vaccinations, limited medical resources.

Divergence-Free Randomized Neural Networks for Solving Incompressible Magnetohydrodynamics Equations

Fei Wang Xi`an Jiaotong University Peoples Rep of China Co-Author(s):    Yunlong Li

Abstract: The incompressible magnetohydrodynamic (MHD) equations are extensively utilized in scientific and engineering fields, yet their strong nonlinearity and two divergence-free conditions pose significant challenges for conventional numerical methods. In this study, we introduce an automatically and precisely divergence-free approach based on Randomized Neural Networks. This method avoids solving nonconvex and nonlinear optimization problems during training, maintains divergence-free properties naturally, and operates as a space-time method. Our proposed approach, named divergence-free randomized neural networks with finite difference method (DF-RNN-FDM), linearizes equations through Picard or Newton iterations, discretizes the problem into a linear system at randomly sampled points across the domain and boundary using the finite difference scheme, and then solves it via a least-square method. We apply this method to solve NS equations, Maxwell equations, and MHD equations. The effectiveness of DF-RNN-FDM is demonstrated by comparison with conventional numerical methods and other neural network-based methods. Our approach achieves higher accuracy with fewer degrees of freedom, simplifies the training process, and precisely adheres to the divergence-free conditions.

Well-posedness and existence of pullback attractors for a class of differential quasi-variational-hemivariational inequalities

Shengda Zeng Chongqing Normal University Peoples Rep of China Co-Author(s):    Shengda Zeng

Abstract: This talk is devoted to introduce our a new research concenring the existence of pullback attractors for a class of differential quasi-variational-hemivariational inequalities involving history-dependent operators. It contains two main results. The first result is to show the existence and uniqueness of solution for the differential quasi-variational-hemivariational inequality under consideration. However, the second one is to develop a new framework for determinating the the existence of the D-pullback attractor of the considered inequality.